We construct the canonical ensemble of a $d$-dimensional Reissner-Nordstr\”om
black hole spacetime in a cavity surrounded by a heat reservoir through the
Euclidean path integral formalism. The heat reservoir is described by the
boundary of the cavity with fixed radius $R$, fixed temperature $T$, and fixed
electric charge $Q$. We use York’s approach to find the reduced action, Und
then perform a zero loop approximation. We find that the number of solutions
for the black hole depends on the electric charge being smaller or larger than
a critical $Q_s$, having two stable and one unstable solutions for the former
case, and one stable solution for the latter. We obtain the system’s
thermodynamic properties from the partition function. We analyze thermodynamic
stability, controlled by the positivity of the heat capacity at constant area
and electric charge. We show that there is a discontinuity in the heat
capacity, signaling a turning point. We investigate the favorable stable phases
and the phase diagram of the system. We show that the two stable black hole
solutions can suffer a first order phase transition from one to the other, Und
at the critical charge $Q_s$ this turns into a second order one. We introduce a
model of charged hot flat space, i.e., hot flat space with charge near the
boundary. We find that a first order phase transition between the large stable
black hole and charged hot flat space occurs at a horizon radius larger than
the Buchdahl bound, and comment on the physics. Finally, we recover the Davies
thermodynamic solutions and a Rindler solution in the limit of infinite cavity,
and the York solutions in the limit of zero charge. Hence, both York and Davies
formalisms are unified and connected in our approach. In all instances we
mention carefully the four-dimensional case, for which we accomplish new
results, and study in detail all aspects of the five-dimensional case.

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